Burns, these are nasty when you are in the city with an emergency room readily available. In the wild, it can be deadly.
The first thing for burns, is to not get one. Sounds simple, doesn't it? When making a fire, take precautions about what type of fuel you are using, and when you add it. After there is a spark, do not put any liquid flammables, like kerosene, on the fire. When using a candle or flame oriented lantern, make sure there is enough ventilation, and at least a foot of clear space around it. And do not have ANYTHING above a flame less than 3 feet, preferably more. Put log stops/rocks around the fire, even if it is a dug pit, to keep burning things from rolling out. When cooking, do not have loose clothing flopping about, and tie up any long hair that could fall into the flame. When putting a fire out, water is good, but using dirt, sand and other dry inert items, and stirring the coals, can snuff out the fire, too. Burry the firepit and put the area back as much like you found it(or better) before you leave.
Check out the How do I make a fire? in the FAQ for a more complete article on starting fires.
If you do have a burn, treat it immediately. Do not wait, as the heat in your skin and tissues will continue to worsen the burn for minutes, even hours, after the inital burn. Water, cold is best, works well for first and second degree burns. Do not use oil based treatments, like butter, until after you have cooled the area. If there is blistering, let it blister - do not pop the blister. It is the body's way of protecting the area.
Cool what you have burned. Water, ice, gel pac. Wet fabric on a first or second degree burn will cool and stop the burn. I do not know if wet or dry fabric would be good on third degree burns - I have read that plastic is better for third degree burns and then cool it. There is no skin when there is a third degree burn.
In the Camping List, I mentioned in the First Aide area to pack Black Tea bags. The tannin in tea can help to stop the burning and start the healing of a burn, especially sunburns. Other things that can work, if you have them available is juiced Aloe(typically Aloe Vera), Oak leaves made into tea(do not drink) and even milk. A poultice made with found herbs can be made, and I recommendPlantain as the main herb. Soak the effected area in what you have, and then wrap the area loosely with as clean of a bandage or cloth as you have.
If you have asprin and can take it, do so. For sunburns and first degree burns, asprin will stop the pain and a lot of the heat damage. For second degree burns, asprin still will work. With third degree burns, asprin can actually harm you - the anti-cloting effects can allow the exposed capillaries to ooze freely and let you bleed enough to make you really weak.
Get to a place where you can get better first aide if the burn breaks the skin at all. Infections are the worst thing with burns. The skin isn't there to keep out things, and so the fat and the muscles below the skin can get really badly infected in a short amount of time.
If it is a third degree burn, do not try to abrade, or clean, the area. That can be worse than leaving it. Get to somewhere that can help you, NOW.